1
SCHEDULE FOR MATHEMATICS SPRING CAMP 2023
START DATE: 30/09 – 06/10/2023
DAYS
CONTENT
ARRIVAL PRE-TEST (STUDY)
DAY
DAY 1
Analytical geometry
DAY 2
Analytical geometry
DAY 3
Functions & Inverses
DAY 4
First principles
and differentiation
rules
ASSESSMENT
PLAN
DATE
SIGNATUR
E
Informal
assessment
Informal
activities.
Informal
activities.
Informal
activities.
Cubic functions
DAY 5
Cubic functions
POST-TEST
(STUDY)
Informal
activities.
DAY 6
TRIGONOMETR
Y
Informal
activities.
DAY 7
PROBABILITY &
COUNTING
PRINCIPLES
Informal
activities.
FEEDBACK
2
ANALYTICAL GEOMETRY
Activity 1(a)
3
ACTIVITY 1(b)
4
Activity 2(a)
Activity 2(b)
5
6
Activity 2(c)
7
Activity 2(d)
8
FUNCTIONS AND INVERSE FUNCTIONS
Activity 3(a)
9
Activity 3(b)
10
11
12
13
Activity 3(c)
4.2
Consider the function h( x) =  1 
x
 3
4.2.1
Is h an increasing or decreasing function? Give a reason for your answer.
4.2.2
Determine h ( x) in the form y = …
(2)
4.2.3
Write down the equation of the asymptote of h(x) – 5.
(1)
4.2.4
Describe the transformation from h to g if g ( x) = log 3 x .
(2)
−1
(2)
14
15
16
ACTIVITY 3(e)
QUESTION 1
1.1
Determine 𝑓 ′ (𝑥) from first principles if 𝑓(𝑥) = 2𝑥 2 + 1
1.2
Determine:
1.2.1
𝑓 ′ (𝑥) if 𝑓(𝑥) = 4𝑥 5 + 6𝑥 4 − 8𝜋
1.2.2
𝐷𝑥 [−4𝑥 8 + ( √𝑥) ]
3
2
(5)
(3)
(3)
QUESTION 2
2.1
Determine the coordinates of A and B
(6)
2.2
Determine 𝑥 −coordinates of the point of inflation of 𝑓.
(2)
2.3
Determine the equation of the tangent of 𝑓 at 𝑥 = 2 in the form 𝑦 = 𝑚𝑥 + 𝑐.
(5)
2.4
Determine the value(s) of 𝑘 for which 𝑓(𝑥) − 𝑘 = 0 will have ONE positive
real root.
(2)
17
QUESTION 3
3.1
3.2
Calculate the:
3.1.1
coordinates of M.
(2)
3.1.2
coordinates of C.
(3)
Determine, giving reasons, the equation of the tangent AB in the form
1
𝑦 = 𝑚𝑥 + 𝑐. If it is given that the gradient of MC is − 2.
(4)
3.3
Calculate the area of ∆𝐴𝐵𝐶.
(5)
3.4
Determine for which value(s) of 𝑘 the line 𝑦 = 2𝑥 + 𝑘 will intersect the circle
at two points.
(5)
18
FIRST PRINCIPLES AND DIFFERENTIATION RULES
Activity 4
7.1
7.2
Determine 𝑓 ′ (𝑥) from first principles if 𝑓(𝑥) = 𝑥 2 − 𝑥
Determine:
𝑓 ′ (𝑥) if 𝑓(𝑥) = 2𝑥 5 − 3𝑥 4 + 8𝑥
7.2.1
1 2
√𝑥
𝐷𝑥 [−
+( ) ]
2
3𝑥
3
7.2.2
7.1
7.2
Determine 𝑓 ′ (𝑥) from first principles if 𝑓(𝑥) = 2𝑥 2 + 1
Determine:
𝑓 ′ (𝑥) if 𝑓(𝑥) = 4𝑥 5 + 6𝑥 4 − 8
7.2.1
7.2.2
7.1
7.2
3
Determine 𝑓 ′ (𝑥) from first principles if 𝑓(𝑥) = −𝑥 2
Given: 𝑦 = 𝑎𝑥 2 + 𝑎 Determine:
𝑑𝑦
𝑑𝑥
7.2.1
𝑑𝑦
𝑑𝑎
7.2.2
7.1
7.2
2
𝐷𝑥 [−4𝑥 8 + ( √𝑥) ]
Determine 𝑓 ′ (𝑥) from first principles if 𝑓(𝑥) = −𝑥 2 + 4
Determine:
19
𝑥3 − 1
𝐷𝑥 (
)
𝑥−1
7.2.1
7.2.2
𝑑𝑦
1 2
if 𝑦 = (𝑥 2 − 𝑥 2 )
𝑑𝑥
CUBIC FUNCTIONS
Activity 5 (a)
20
Activity 5(b)
21
Activity 5(c)
22
23
24
TRIGONOMETRY
QUESTION 5
In the diagram below, P (–15 ; m) is a point in the third quadrant and 17cos β + 15 = 0.
y
β
O
x
.
P (–15 ; m)
25
5.1
WITHOUT USING A CALCULATOR, determine the value of the following:
5.1.1 m
5.2
(3)
5.1.2
sin β + tan β
(3)
5.1.3
cos 2β
(3)
Simplify:
sin(180 − x).cos(x − 180 ). tan(360 − x)
sin(− x).cos(450 + x)
5.3
3.1
3.2
Consider the identity:
(7)
sin x + sin 2 x
= tan x
1+ cos x + cos 2 x
5.3.1
Prove the identity.
(5)
5.3.2
Determine the values of x for which this identity is undefined.
(4)
[25]
If sin 31° = p, determine the following, without using a calculator, in terms of p:
3.1.1
sin 149°
(2)
3.1.2
cos (–59°)
(2)
3.1.3
cos 62°
(2)
Simplify the following expression to a single trigonometric ratio:
tan(180 −  ).sin 2 (90 +  ) + cos( − 180).sin 
(6)
26
QUESTION 5
In the diagram below, CGFB and CGHD are fixed walls that are rectangular in shape and vertical
to the horizontal plane FGH. Steel poles erected along FB and HD extend to A and E
respectively. ∆ACE forms the roof of an entertainment centre.
BC = x, CD = x + 2, BÂC =  , AĈE = 2 and EĈD = 60
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E
A

D
2
B
x
60°
x+2
C
H
F
G
5.1
Calculate the length of:
5.1.1
AC in terms of x and 
(3)
5.1.2
CE in terms of x
(2)
5.2
Show that the area of the roof ACE is given by 2 x ( x + 2) cos .
(4)
5.3
If  = 55 and BC = 12 metres, calculate the length of AE.
(4)
[13]
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PROBABILITY & COUNTING PRINCIPLES
QUESTION 9 ( Probablity)
The success rate of the Fana soccer team depends on a number of factors. The fitness of the players
is one of the factors that influence the outcome of the match.
• The probability that all the players are fit for the next match is 70%
• If all the players are fit to play the next match, the probability of winning the next match
is 85%
• If there are players that are not fit to play the next match, the probability of winning the
match is 55%
9.1
9.2
Draw a tree diagram and calculate the probability that the Fana soccer team will
win the next match.
C and D are two events where:
(4)
3
1
4
P(C) = , P ( D ) = , P ( C or D) =
5
3
15
9.2.1
9.2.2
Determine the P(C and D).
Are C and D independent events ? Show all your calculations
(4)
(3)
[11]
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30
31
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